Real grain shape analysis: characterization and generation of representative virtual grains. application to railway ballast

545 Real g ain shape analysis: cha ac e iza ion and gene a ion o ep esen a i e i ual g ains. Applica ion o ailway ballas IV In e na ional Con e ence on Pa icle-based Me hods – Fundamen als and Applica ions PARTICLES 2015 E. Oña e, M. Bischo , D.R.J. Owen, P. W igge s & T. Zohdi (Eds) REAL GRAIN SHAPE ANALYSIS: CHARACTERIZATION AND GENERATION OF REPRESENTATIVE VIRTUAL GRAINS. APPLICATION TO RAILWAY BALLAST. N. OUHBI¹2, C. VOIVRET2, G. PERRIN3 AND J-N. ROUX1 ¹ Labo a oi e Na ie (UMR CNRS 8205) CNRS, Ecole des Pon s e Chaussées, IFSTTAR Champs-su -Ma ne, F ance e-mail: {jean-noel. oux}@i s a . ²Inno a ion and Resea ch Depa men , SNCF, Pa is, F ance Immeuble Lumiè e 40 A enue des Te oi s de F ance, F-75611 PARIS CEDEX 12 e-mail: {nou a.ouhbi, cha les. oi e }@snc . ³CEA/DAM/DIF F-91297, A pajon, F ance Keywo ds: Ballas , pa icle shape, Cha ac e iza ion, DEM, polyhed a, POD, gene a ion. Abs ac . G ain shape signi ican ly in luences he mechanical p ope ies o g anula media. In o de o explo e his e ec and o simula e ealis ic ma e ial mo phology, we designed a me hod which well cha ac e izes eal g ains shape. S a ing om a ep esen a ion o he pa icle su aces as a poin s cloud, his pape p esen s a me hod o gene a e a se o i ual g ains ha a e mo phologically ep esen a i e o eal ballas g ains. The model elies on a s a is ical modelling o he ballas g ain mo phology based on a dimensionali y educ ion app oach (P ope O hogonal Decomposi ion) leading o an op imal and nea ly exhaus i e shape cha ac e iza ion by ex ac ing a hie a chy o shape unc ions ha ully desc ibe he g ain sample. We will show he e iciency o he bo h cha ac e izing and gene a ing me hods and desc ibe hei ad an ages, as well as a u u e ou look 1 INTRODUCTION G anula ma e ials a e widely used in di e en applica ions anging om ood indus y o ci il enginee ing. The e o e, a be e unde s anding o he o e all beha iou o hese ma e ials is pi o al o imp o e and con ol hei pe o mance. Nume ous s udies ha e been ca ied ou and en iched o e he las yea s, abou he impac o pa icle size, shape and mine alogy on he mechanical beha iou o he g anula media. As o shape p ope ies, se e al expe imen al analyses as well as nume ical s udies using Disc e e Elemen Me hods (DEM) [1] ha e shown he signi ican in luence o pa icle shape on he e olu ion o g anula assemblies [2-6, 11-13, 16, 17, 24, 26]. Unde s anding his in luence is hen a opic o in e es in his s udy. 546 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 2 Fo DEM simula ions, ep esen ing pa icle shape is a eal challenge. Mos disc e e elemen codes ep esen pa icles as discs (2D) o sphe es (3D), because o implemen a ion simplici y and compu a ional e iciency. Howe e , i has been p o ed ha hese simple geome ies ail o ep oduce ealis ic ma e ial beha iou o an indi idual g ain, as well as o a g anula assembly [7, 14, 20]. O he a emp s ha e hen been made o ake in o accoun complexi y ha alls oughly in wo classes: 1) pa ame ic model based on gi en geome ic cons uc ion ules and 2) i ed model based on eal g ain shapes. Bo h app oaches can be ul illed wi h simple p imi i e (sphe e, ellipsoid, plane, clumps...) o wi h mo e e sa ile geome ic shape as polyhed on [8, 10]. O e he las decades, SNCF (F ench ailway company) has chosen he second app oach o simula e he beha iou o he ballas wi h DEM based on polyhed on shaped pa icles. Railway ballas is a g anula laye o med by i egula ock g ains o a cen ime ic size ex ac ed om ha d s one qua ies by c ushing ((BS EN 13450, 2003) [1]). Cu en ly, i ual g ains used in simula ions a e se s o nea ly 1000 sampled g ains ha ha e been 3D digi alized and meshed. In o de o p ope ly s udy he impac o he ballas shape on he mechanical beha iou wi h DEM, a gene a o o i ual g ains is needed, ap o p oduce la ge se s o i ual g ains ha a e ep esen a i e o a limi ed se o eal g ain, and ha also allows o each an accu a e cha ac e iza ion o he g ain shape. In his pape , we p opose a me hod o achie e hese goals. In Sec ion 2, he p oposed app oach o eal ballas g ain shape modelling as well as i s alida ion a e p esen ed and analysed. Some illus a ions a e p esen ed in Sec ion 3. 2 REAL BALLAST GRAINS MODELLING 2.1 Global iew In he li e a u e, gene a ion me hods we e p esen ed, such as Fou ie -Shape-Desc ip o s [19, 22], and sphe ical-based andom ields [9, 18, 29]. These app oaches ha e gi en good esul s in e ms o simila i y o gene a ed g ains and eal ones, bu in oduce shape unc ions ha a e imposed. We p esen in his pape an inno a i e app oach based on a dimensionali y educ ion me hod leading o an op imal and nea ly exhaus i e shape cha ac e iza ion o eal g ains. By means o P ope O hogonal Decomposi ion (POD) [15], we iden i y he op imal hie a chy o shape unc ions ha desc ibe he g ain se . The main ad an age o his app oach is o educe he numbe o needed shape unc ions o ep esen he g ain shape wi h a quan i a i e con olled app oxima ion (e o based), such ha we educe he pa ame ic space o he op imal one. 2.2 P e-p ocessing: Sample p epa a ion A da abase o 121 di e en ballas g ains is p o ided by SNCF. Ballas ma e ials in F ance a e selec ed based on he Eu opean ailway ballas speci ica ion. The g ains a e ep esen ed by poin clouds ob ained expe imen ally by 3D digi iza ion (3D scan) o he pa icle su aces, and o m i egula polyhed ons o 4000 aces, and abou 2000 e ices. Fig.1 shows some o hese g ains. 547 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 3 Figu e 2: A g ain o ballas : (a) Raw da a (b) A e p ojec ion and o a ion The p e-p ocessing is based on he ollowing s eps:  P ojec ion o he expe imen al poin s cloud on an imposed basis (B1) wi h cons an angula s ep and N di ec ions. Since he expe imen al poin cloud a e composed o many poin s (up o millions o ine digi aliza ion) wi h andom di ec ion in a sphe ical ame, his s ep aims a educing he numbe o poin and ha ing he same di ec ion o each g ain.  Compu a ion o all g ains o olume, su ace, ine ia enso , and mean adius.  Cen e ing g ains and o a ing e ices un il he p incipal di ec ions o he ine ia enso a e pa allel o he global coo dina e axes, and in e pola ing (Fig.2)  P ojec ion on a new basis (B2) wi h uni o m densi y o numbe o di ec ion by solid angle. Indeed, he p e ious basis, wi h cons an angula s ep, lead o a high densi y o di ec ion nea he pole ha can bias he s a is ic. This new basis wi h 500 di ec ions is gene a ed wi h a epulsion poin i e a i e p ocedu e. (Fig. 3)  By conca ena ing all g ains oge he , ou se o expe imen al g ain is hen ep esen ed by a 500 x 121 ma ix. Each column ep esen s g ain e ices dis ances along he 500 o he basis B2 (Fig.4). This ma ix will be he inpu da a o POD p ocedu e. Figu e 1 : Example o eal ballas g ains ep esen ed by dense poin clouds - SNCF (a) (b) 548 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 4 Figu e 4: In e pola ion on he gene a ed basis wi h 500 di ec ions and eal g ain 2.3 G ain cha ac e iza ion A model educ ion me hod, namely he P ope O hogonal Decomposi ion o P incipal Componen Analysis (PCA) [27] is used. S a ing om he da a ma ix o he Sec ion 2.2, his mul i a ia e s a is ical me hod aims a ob aining a compac ep esen a ion o he da a. I iden i ies an op imal and use ul se o basic unc ions ha allows o achie e a sa is ac o y app oxima ion o he sys em. This me hod se es wo pu poses, namely o de educ ion by p ojec ing high-dimensional da a in o a lowe -dimensional space and ea u e ex ac ion by e ealing ele an , bu unexpec ed, s uc u e hidden in he da a. The i s pu pose allows o quan i y he con olled app oxima ion (e o based), while choosing he dimension o he p ojec ion educed space. The second pu pose allows o cha ac e ize he shape ea u es by associa ing hem o POD ou pu s, i.e. basis unc ions (eigenmodes) and coe icien s. Eigenmodes, o p incipal componen s, a e he eigen ec o s o he co a iance ma ix co esponding o o iginal da a, whe eas coe icien s a e p ojec ion coo dina es on he educed space o p ojec ion. These wo elemen s, in addi ion o eigen alues o he co a iance ma ix a e he h ee key elemen s o he me hod. Fo u he explica ions o he me hod, see [15]. While applying POD on he da a, a co esponding e o ε is exp essed in e ms o he p ojec ion e o s ha a e con olled in he cons uc ion o POD bases. In o he wo ds, i is de ined as he de ia ion o he ans o med da a o he new space om he o iginal da a, no malized by he aw da a and induced by he unca ion o he POD basis. While POD e o gi es an indica ion o he magni ude o he “missing” in o ma ion, he Figu e 3: Basis wi h 500 di ec ion: (a) basis B1 (b) basis B2 (a) (b) 549 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 5 ene gy o he sys em 𝑒𝑒(𝑘𝑘) = 1 − 𝜀𝜀(𝑘𝑘) ep esen s he quan i y o in o ma ion cap u ed by he k i s POD basis ec o s. (Fig. 5) shows he quan i ies o e o /ene gy as a unc ion o he dimension o he educed space. Fo ze o e o we ha e o keep all 121 modes. We see ha wi h only 12 modes we ep esen 90% o he in o ma ion and 99% o he in o ma ion is ep esen ed wi h oughly 50 modes. Wi h his p ecision, we ha e hal - educe ou da a. As he accu acy o he g ain app oxima ion depends on he numbe o modes, i is impo an o check i he p ope ies o g ains (a e age adius, su ace, olume) a e sensi i e o he numbe o modes. In (Fig.6), we compa e hose p ope ies dis ibu ions o di e en e o h esholds. In o de o quan i y mo e p ecisely he compa isons, we pe o m Kolmogo o -Smi no (KS) es s be ween o iginal da a su aces/ olumes/mean adius and econs uc ed g ains cha ac e is ics. The esul s a e p esen ed in Table 1. Table 1: Values o Kolmogo o -Smi no es s be ween o iginal and econs uc ed da a S, V, Rm dis ibu ions E o on shape KS Su aces KS Volumes KS Rayons moyens 0.1% 0.999 0.997 0.947 1% 0.999 0.997 0.879 5% 0.999 0.997 0.785 10% 0.997 0.984 0.465 15% 0.984 0.879 0.223 Figu e 5: E o and ene gy quan i ica ion 550 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 6 We see ha e en wi h signi ican e o (mo e han 10%) we ha e a e y good shape app oxima ion bu wi h a much mo e educed space. Fu he mo e, by s a is ically iden i ying POD coe icien s dis ibu ions, we can gene a e as many equi alen da a as wan ed. Fo a pe ec ly exhaus i e shape cha ac e iza ion, we decide o keep all o he eigenmodes and he e o e an e o o 0% is made. The s a is ical analysis will conce n he coe icien s ha a e unco ela ed. (Fig. 7) shows s a is ical dis ibu ions o he i s 8 coe icien s as an example. Figu e 6: Recons uc ion o he eal g ains o di e en e o h esholds – Compa ison o (a) Su aces dis ibu ions (b) Volumes dis ibu ions (c) Mean adius dis bu ions Figu e 7: S a is ic ep esen a ion o he i s 8 coe icien s Expe imen al da a ε = 0.1% ε = 1% ε = 5% ε = 10% ε = 15% Expe imen al da a ε = 0.1% ε = 1% ε = 5% ε = 10% ε = 15% Expe imen al da a ε = 0.1% ε = 1% ε = 5% ε = 10% ε = 15% 551 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 7 The challenge is hen o build a model o each dis ibu ion. While i is possible o sea ch he bes ma ching dis ibu ion o o i he CDF, we p esen ly p e e o keep he exac CDFs co esponding o disc e e da a, wi hou ans o ming i o con inuous i ed unc ions (Fig. 8). Figu e 8: Fi s coe icien CDF 2.4 Gene a ion E en hough he coe icien s a e unco ela ed (no shown), hese coe icien s can be dependen in a mo e complex way. The e o e, he gene a ion p ocess has o ake in o accoun he dependence. Since he coe icien s a e o di e en ma ginal dis ibu ions, his canno be done by only using a mul i a ia e dis ibu ion, which su e s om he es ic ion ha he ma ginal should be o he same ype. One e icien way o do his is copula unc ions [28]. Tha is, copula unc ions allow one o model he dependence s uc u e independen ly o he ma ginal dis ibu ions. Any mul i a ia e dis ibu ion unc ion can se e as a copula. Tha o e s a good modelling lexibili y. Fo ou gene a ion p ocess, we use he Gaussian copula, since wo pa ame e s (Mean and co ela ion ma ices) a e enough o desc ibe i . The algo i hm is explained in de ails in [21, 22, 25]. 2.5 Valida ion To alida e ou app oach, we gene a e di e en se s o 300, 500, 800, 1000, 2000 and 10000 i ual g ains. Su aces, olumes and mean adius dis ibu ions a e hen compu ed and compa ed o hose o he o iginal da a. (Fig. 9) shows he esul s, and Kolmogo o -Smi no es s esul s a e p esen ed in Table 2. Expe imen al CDF Fi ed CDF 552 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 8 Table 2: Values o Kolmogo o -Smi no es s be ween o iginal gene a ed da a S, V, Rm dis ibu ions By analysing Fig. 9, we see ha o su aces, olumes and a e age adii, he peaks a e he poin s ha ma ch he leas . Tha can be explained by he ac ha he eal pa icle da ase (121 g ains) is ela i ely small, and insu icien o es ima ing accu a ely s a is ics o he g ains shape, and hence cap u ing all shape ea u es. Tha was p o ed by Liu e al. [18] who analysed he impac o da a se size on he accu acy o esul s using sphe ical ha monics app oach, and by G igo iu e al. [9] who also p o ed he insu iciency o a da ase o 128 Numbe o gene a ed g ains KS Su aces KS Volumes KS Rayons moyens 300 0.949 0.949 0.811 500 0.993 0.993 0.831 800 0.994 0.994 0.883 1000 0.998 0.939 0.705 2000 0.976 0.896 0.870 10000 0.939 0.968 0.613 Figu e 9: Sample alida ion – 10000 g ains gene a ed Expe imen al da a Gene a ed da a Expe imen al da a Gene a ed da a Expe imen al da a Gene a ed da a Expe imen al da a Gene a ed da a 553 N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux 9 agg ega es o exac ly ep esen he conc e e g ains shape. They showed ha a “small da a se was unable o accu a ely es ima e s a is ics o agg ega e geome y beyond second-momen p ope ies and ma ginal dis ibu ion”. Howe e , he high alues o KS es s ob ained and he ep esen a ions show ha we ge sa is ac o y esul s, e en wi h a small da a se . Tha con i ms he accu acy o he app oach and shows ha en iching ou da a se will ce ainly allow o ge an e en be e and exhaus i e cha ac e iza ion o ballas shape, and o e an in e es ing ansi ion be ween eal and i ual g ains in o de o inco po a e hem in DEM simula ions. 3 ILLUSTRATION As shown p e iously, POD p ocedu e allows o e y well app oxima ing he g ains shape h ough eigenmodes and coe icien s. One in e es ing ques ion would be how o link eigenmodes and coe icien s o shapes ea u es. One o he ad an ages o his me hod is o de ing he domina ing ea u es by dec easing o de ( i s modes ha e he bigges con ibu ions o ene gy o he sys em, and he la ges alues o a iance). We hen expec he i s mode o hold an impo an pa o shape ea u es (wi h 45% o he ene gy o he sys em). To obse e he e olu ion o g ains shape h ough modes, we ep esen he same g ain econs uc ed wi h he i s 1 (ene gy = 45%) o 20 modes (95%) in Fig. 11. We can see he p og essi e eme gence o shape de ails as we add new modes. As he quan i y o in o ma ion inc eases (e o dec easing), we minimize he sum o he squa ed di e ences o he dis ances be ween he poin on he eal g ain and he same one ep esen ed wi h a ini e numbe o basic unc ions. The i s ou h modes a e shown in Fig.12. We see ha he i s mode, con ibu ing mos o he ene gy o he sys em, has a comple e shape o a g ain while he es o he modes cons i u e he o he de ails o shape. Finally, we ep esen econs uc ed g ains wi h only he i s mode in Fig. 13. Fo all 121 g ains, we ha e a “ ound” shape. Figu e 11: G ain shape e olu ion h ough modes 1 o 20 and eal g ain